A Functional Wavelet-Kernel Approach for Continuous-time Prediction

TL;DR

This paper introduces a wavelet-kernel nonparametric regression method for predicting continuous-time stochastic processes using segmented functional data, with proven asymptotic properties and practical validation on real datasets.

Contribution

It develops a novel wavelet-based kernel approach for continuous-time prediction from inhomogeneous functional data, including asymptotic analysis and confidence interval construction.

Findings

Effective prediction of continuous-time processes demonstrated on real datasets.

Asymptotic properties established under mild conditions.

Nonparametric confidence intervals provide reliable uncertainty quantification.

Abstract

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on functional kernel nonparametric regression estimation techniques where observations are segments of the observed process considered as curves. These curves are assumed to lie within a space of possibly inhomogeneous functions, and the discretized times series dataset consists of a relatively small, compared to the number of segments, number of measurements made at regular times. We thus consider only the case where an asymptotically non-increasing number of measurements is available for each portion of the times series. We estimate conditional expectations using appropriate wavelet decompositions of the segmented sample paths. A notion of similarity, based on wavelet decompositions, is used in order to calibrate the prediction. Asymptotic properties when the number of segments grows to infinity are investigated under mild conditions, and a nonparametric resampling procedure is used to generate, in a flexible way, valid asymptotic pointwise confidence intervals for the predicted trajectories. We illustrate the usefulness of the proposed functional wavelet-kernel methodology in finite sample situations by means of three real-life datasets that were collected from different arenas.